PI: Vera Guelpers
The Standard Model (SM) of particle physics is our best current description of the fundamental building blocks of the universe. The SM is consistent with most experimental observations, but has a number of shortcomings necessitating New Physics. One such discrepancy is that neutrinos are massless in the SM, but have been experimentally observed to have finite mass.
As a result the nature of neutrinos poses a big open question in theoretical physics, in particular whether they are Dirac or Majorana fermions. In the latter case they are their own antiparticle, which would allow some decays which are otherwise forbidden, including neutrinoless double beta decay, the transition nn →p+p+ e–e– with no neutrinos emitted and which violates lepton number conservation. This can in turn lead to baryon number violation which would help explain the observed matter-antimatter asymmetry of the universe. To date this process has not been experimentally observed, but several experimental searches are underway.
Theory predictions are vital to interpret the experimental results. One important input is the pion exchange contribution π–→π+e–e– which we calculate in this work. This is achieved by computing the matrix elements of five distinct four-quark operators between pions. These matrix elements are governed by the quantum field theory of the strong force (Quantum Chromodynamics or QCD) at low energies where traditional perturbative methods converge poorly. Instead, the method of choice for controlled calculations is lattice QCD, an ab initio method that starts directly from the QCD Lagrangian by numerical evaluation of the path integral.
This work provides a milestone result, addressing tensions between two previous calculations and reducing several sources of uncertainty. First, we compute directly at physical quark masses, eliminating the need for an extrapolation to the physical pion mass. Second, we devised a method to significantly reduce “around-the-world” effects, that is artifacts stemming from the finite temporal extent of the simulation. Since the quantities of interest are extracted from correlation functions that decay exponentially in time, this is a major improvement as it largely removes the need to parameterise these effects and hence extends the signal region, making better use of the generated data. Third, we use chirally-symmetric domain wall fermions and compute non-perturbative renormalisation constants in two different schemes further reducing systematic uncertainties.
With respect to the tension between the two previous results we observe the following: depending on the operator we differ by 2 to 6σ from the CalLat result by Nicholson et al., and when comparing with Detmold et al. our results differ by a factor compatible with two for each of the five matrix elements. This hints at a normalisation discrepancy and we find good agreement if we account for this factor. Exploiting the relation to neutral kaon mixing and evaluating the equivalent matrix elements for pions, we agree with the literature values for both. This non-trivial cross check makes us confident in our normalisation. The figure below shows results for the five matrix elements <Oi>, together with our cross check labelled by Bπ.
Depending on the matrix element, we achieve a relative precision of 1%-7%, improving on existing results in all cases, in some cases by a factor of 5. Future improvement of the lattice uncertainties could be achieved through increased statistical sampling for evaluating the path integral, the inclusion of finer lattice spacings and larger volumes. However, as shown in the figure, for most operators the precision is limited by the truncation of the perturbatively computed matching factors needed to convert the lattice results to standard conventions.
References:
P. Boyle, F. Erben, X. Feng, J. Flynn, N. Garron, T. Izubuchi, L. Jin, R. Mukherjee, J.T.Tsang, X-Y. Tuo arXiv:2508.01900